Relaxation time and topology in 1D $O(N)$ models

TL;DR

This paper investigates how the topology of the state space in 1D $O(N)$ models influences their relaxation times, revealing a link between topological features and exponential or polynomial temperature dependence.

Contribution

It demonstrates that the relaxation time in 1D $O(N)$ models reflects the model's topological properties, depending on boundary conditions and homotopy classes.

Findings

Metastable states cause exponential relaxation times at low temperature.

Topology determines whether relaxation time is exponential or polynomial.

The relaxation time serves as a proxy for the model's topological features.

Abstract

We discuss the relaxation time (inverse spectral gap) of the one dimensional $O(N)$ model, for all $N$ and with two types of boundary conditions. We see how its low temperature asymptotic behavior is affected by the topology. The combination of the space dimension, which here is always 1, the boundary condition (free or periodic), and the spin state $S^{N-1}$, determines the existence or absence of non-trivial homotopy classes in some discrete version. Such non-trivial topology reflects in bottlenecks of the dynamics, creating metastable states that the system exits at exponential times; while when only one homotopy class exists the relaxation time depends polynomially on the temperature. We prove in the one dimensional case that, indeed, the relaxation time is a proxy to the model's topological properties via the exponential/polynomial dependence on the temperature.